2. Games People Play. 10: Evolutionary Games

3. Games People Play. Evolutionary Games What if individuals aren’t as smart and calculating as we have assumed so far? Perhaps decision making is simpler Good decisions and decision makers persist and are copied Bad decisions and decision makers die out!! Natural selection makes the decisions This is the idea behind evolutionary game theory

4. Games People Play. Evolutionary Games Genotype – the genetic type of a player Phenotype – the behavior of a genotype Fitness – a measure of the success of a phenotype Selection – successful genotypes out-reproduce unsuccessful ones Mutations – random creation of new genotypes Invasion – mutations that successfully out-compete the current genotypes and increase in number Evolutionary stability – a population of genotypes that cannot be successfully invaded

5. Games People Play. The prisoners dilemma as an evolutionary game 2 genotypes Cooperators (C-types) – always cooperate Defectors (D-types) - always defect Pairs of players are matched at random A cooperator can be matched with another cooperator or with a defector and vice versa.

6. Games People Play. The prisoners dilemma as an evolutionary game The payoff matrix Suppose that the proportions of cooperators and defectors in the population are initially x and 1-x respectively. Column CooperateDefect Row Cooperate 12,12 1, 25 Defect 25,1 3,3

7. Games People Play. The prisoners dilemma as an evolutionary game Fitness levels A cooperator meets another cooperator with probability x and a defector with probability 1-x and expects to earn F(c) = x(12) + (1-x)(1) A defector will also meet a cooperator with probability x and a defector with probability 1-x and expects to earn F(d) = x(25) + (1-x)(3) Column CooperateDefect Row Cooperate 12,12 1, 25 Defect 25,1 3,3

8. Games People Play. The prisoners dilemma as an evolutionary game Selection The cooperators will outbreed the defectors if F(C) > F(D) x(12) + (1-x)(1) > x(25) + (1-x)(3) So the cooperators will outbreed the defectors if x < - (2/11) Which cannot hold. So the cooperators will die out!!! 100% defectors is an Evolutionary Stable State.

9. Games People Play. The prisoners dilemma as an evolutionary game Mutation Suppose now a mutation occurs and a third genotype that plays tit-for-tat appears. Also suppose that each pair of players plays each other three times. Assume a T-type always plays cooperate on the first round. Can the mutation successfully invade?

10. Games People Play. The prisoners dilemma as an evolutionary game If a defector meets a defector we get Round #1 Round #2 Round #3 Defector CooperateDefect Defector Cooperate 12,12 1,25 Defect 25,1 3,3 DefectorCooperateDefect DefectorCooperate 12,12 1,25 Defect 25,1 3,3 DefectorCooperateDefect DefectorCooperate 12,12 1,25 Defect 25,1 3,3

11. Games People Play. The prisoners dilemma as an evolutionary game So each defector that meets another defector enjoys a fitness of 9

12. Games People Play. The prisoners dilemma as an evolutionary game If a defector meets a tit- for-tat we get Round #1 Round #2 Round #3 Defector CooperateDefect Tit-for- tat Cooperate 12,12 1,25 Defect 25,1 3,3 DefectorCooperateDefect Tit-for- tat Cooperate 12,12 1,25 Defect 25,1 3,3 DefectorCooperateDefect Tit-for- tat Cooperate 12,12 1,25 Defect 25,1 3,3

13. Games People Play. The prisoners dilemma as an evolutionary game If a defector meets a tit-for-tat The tit-for-tat enjoys a fitness of 7 The Defector enjoys a fitness of 31

14. Games People Play. The prisoners dilemma as an evolutionary game If a tit-for-tat meets a tit- for-tat we get Round #1 Round #2 Round #3 Tit-for-tat CooperateDefect Tit-for- tat Cooperate 12,12 1,25 Defect 25,1 3,3 Tit-for-tatCooperateDefect Tit-for- tat Cooperate 12,12 1,25 Defect 25,1 3,3 Tit-for-tatCooperateDefect Tit-for- tat Cooperate 12,12 1,25 Defect 25,1 3,3

15. Games People Play. The prisoners dilemma as an evolutionary game If a tit-for-tat meets a tit-for-tat Both tit-for-tats enjoy a fitness of 36

16. Games People Play. The prisoners dilemma as an evolutionary game So we know Defector meets defector Both receive a fitness of 9 Defector meets a tit-for-tat Defector receives a fitness of 31 Tit-for-tat receives a fitness of 7 Tit-for-tat meets a tit-for-tat Both receive a fitness of 36.

17. Games People Play. The prisoners dilemma as an evolutionary game Conclusions There are two evolutionary stable steady states A tit-for-tat cannot invade a population of defectors A defector cannot invade a population of tit-for-tats If both types initially exist in the population which ESS arises depends on their initial relative numbers. History matters

18. Rats! Games People Play.

19. Colonel Blotto Game Battlefield Team Name 12345678910 Hold Up Guys 33563123357 3 Window Shoppas 3626 36252 526 Wasted Potential 50510 50010550 Cemetery Scuba Squad 6046343634 12 Sarah Palin Wannabes330341330 0 Zoo0133 0 1 Unemployed Pixies35555 5 5 Conquistadors3211 1331 Dundermifflin290 280 030 Bush Lovers000034 33 Games People Play.

20. Colonel Blotto Game Results Games People Play. HUGWSWPCSSSPWZUPCDBLTotal Hold Up Guys4-65-56-41.5 Window Shoppas6-45-46-43.0 Wasted Potential5-54-56-41.5 Cemetery Scuba Squad4-6 3-60.0 Sarah Palin Wannabes6-34-42-81.5 Zoo4-43-77-31.5 Unemployed Pixies8-27-3 3.0 Conquistadors3-78-24-32.0 Dundermifflin3-72-83-60.0 Bush Lovers4-63-46-31.0